Algebras with Young tableaux theories

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 26 February 2013

Algebras with Young tableaux theories

A. Young invented the theory of standard Young tableaux in order to describe the representation theory of the symmetric group Sn; the group of n×n matrices such that

  1. the entries are either 0 or 1,
  2. there is exactly one nonzero entry in each row and each column.

Young himself began to generalize the theory and in [You1929] he provided a theory for the Weyl groups of type B, i.e. the hyperoctahedral groups WBn(/2)Sn of n×n matrices such that

  1. the entries are either 0 or ±1,
  2. there is exactly one nonzero entry in each row and each column.

In the same paper Young also treated the Weyl group WDn of n×n matrices such that

  1. the entries are either 0 or ±1,
  2. there is exactly one nonzero entry in each row and each column,
  3. the product of the nonzero entries is 1.

W. Specht [Spe1932] generalized the theory to cover the complex reflection groups Gr,1,n (/r)Sn consisting of n×n matrices such that

  1. the entries are either 0 or rth roots of unity,
  2. there is exactly one nonzero entry in each row and each column.

In the classification [STo1954] of finite groups generated by complex reflections there is a single infinite family of groups G(r,p,n) and exactly 34 others, the “exceptional” complex reflection groups. The groups G(r,p,n) are the groups of n×n matrices such that

  1. the entries are either 0 or rth roots of unity,
  2. there is exactly one nonzero entry in each row and each column,
  3. the (r/p)th power of the product of the nonzero entries is 1.

Though we do not know of an early reference which generalizes the theory of Young tableaux to these groups, it is not difficult to see that the method that Young uses for the Weyl groups WDn extends easily to handle the groups G(r,p,n).

Special cases of the groups G(r,p,n) are

  1. G(1,1,n)=Sn, the symmetric group,
  2. G(2,1,n)=WBn, the hyperoctahedral group (i.e. the Weyl group of type Bn),
  3. G(2,2,n)=WDn, the Weyl group of type Dn,
  4. Gr,1,n (/r)Sn =(/r)n Sn.

The order of G(r,1,n) is rnn!. Let Eij be the n×n matrix with 1 in the (i,j) position and all other entries 0. Then G(r,1,n) can be presented by generators

s1=ζE11+ j1Eii, andsi= Ei,i+1+ Ei+1,i+ ji,i+1 Ejj,2 in,

where ζ is a primitive rth root of unity, and relations

(B1) sisj=sjsi, ifi-j>1, (B2) sisi+1si =si+1si si+1, for2in-1, (BB) s1s2s1s2= s2s1s2s1, (C) s1r=1, (R) si2=1, for2in.

The group G(r,p,n) is the subgroup of index p in G(r,1,n) generated by

a0=s1p, a1=s1s2s1, ai=si,2 in.

Cyclotomic Hecke algebras Hr,1,n

More recently there has been an interest in Iwahori-Hecke algebras associated to reflection groups and there has been significant work generalizing the constructions of A. Young to these algebras. Iwahori-Hecke algebras of types A, B and D were handled by Hoefsmit [Hoe1974] and other aspects of the theory for these algebras were developed by Dipper, James and Murphy [DJa1987], [DJM1995], Gyoja [Gyo1986] and Wenzl [Wen1988]. In 1994, Ariki and Koike [AKo1994] introduced cyclotomic Hecke algebras Hr,1,n for the complex reflection groups G(r,1,n) and they generalized the Young tableau theory to these algebras. Theorem 3.18 below shows that the theory of [AKo1994] is a special case of an even more general theory for affine Hecke algebras.

Let u1,,ur,q, q0. The cyclotomic Hecke algebra is the algebra Hr,1,n (u1,,ur;q) , over , given by generators T1,,Tn and relations

(B1) TiTj=TjTi, ifi-j>1, (B2) TiTi+1Ti= Ti+1Ti Ti+1, for2in-1, (BB) T1T2T1T2= T2T1T2T1, (qC) (T1-u1) (T1-u2) (T1-ur)=0. (1R) (Ti-q) (Ti+q-1)=0, for2in.

The algebra Hr,1,n (u1,,ur;q) is of dimension rnn! (see [AKo1994]).

  1. H1,1,n(1;q) is the Iwahori-Hecke algebra of type An-1.
  2. H2,1,n(q,-q-1;q) is the Iwahori-Hecke algebra of type Bn.
  3. If ζ is a primitive rth root of 1 then Hr,1,n (1,ζ,,ζr-1;1) is the group algebra G(r,1,n).

Fact (c) says that the representation theory of the groups G(r,1,n) is a special case of the representation theory of the algebras Hr,1,n.

Cyclotomic Hecke algebras Hr,p,n

The work of Broué, Malle and Michel [BMM1993] demonstrated that there are cyclotomic Hecke algebras associated to most complex reflection groups (even exceptional complex reflection groups). In particular, there are cyclotomic Hecke algebras Hr,p,n corresponding to all the groups G(r,p,n) and Ariki [Ari1995] has generalized the Young tableau mechanism to these groups (see also [HRa1998]). Theorems 3.15 and the mechanism of Theorem 2.8 show that the theory of Ariki is a special case of a general construction for affine Hecke algebras.

Let r,p,n>0 be such that p divides r and let d=r/p. Let x0,,xd-1 and let ξ be a primitive pth root of 1. For 1jr, define

uj=ξkx, ifj-1=p+k, ( 0kp-1,0 d-1 )

i.e., u1,,ur are chosen so that

(T1-u1) (T1-u2) (T1-ur)= (T1p-x0p) (T1p-x1p) (T1p-xd-1p).

The algebra Hr,p,n (x0,,xd-1;q) is the subalgebra of Hr,1,n (u1,,ur;q) generated by the elements

a0=T1p, a1=T1-1 T2T1,ai =Ti,for2 in.

Then

  1. H2,2,n(1;q) is the Iwahori-Hecke algebra of type Dn,
  2. If η is a primitive dth root of unity then Hr,p,n (1,η,,ηd-1;1) is the group algebra G(r,p,n).

Affine braid group of type A

There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:

  1. As braids in a (slightly thickened) cylinder,
  2. As braids in a (slightly thickened) annulus,
  3. As braids with a flagpole.

See Figure 1. The multiplication is by placing one cylinder on top of another, placing one annulus inside another, or placing one flagpole braid on top of another. These are equivalent formulations: an annulus can be made into a cylinder by turning up the edges, and a cylindrical braid can be made into a flagpole braid by putting a flagpole down the middle of the cylinder and pushing the pole over to the left so that the strings begin and end to its right.

The group formed by the affine braids with n strands is the affine braid group of type A. The affine braid group ,1,n is presented by generators T2,,Tn and Xε1 (see Figure 2) with relations

(B1) TiTj=TjTi, ifi-j>1, (B2) TiTi+1Ti= Ti+1Ti Ti+1, for2in-1, (BB) Xε1T2 Xε1T2=T2 Xε1T2 Xε1, (B1) Xε1Ti= TiXε1, for3in.

Inductively define Xεi,1,n by

Xεi=Ti Xεi-1 Ti,2in. (1.4)

By drawing pictures of the corresponding affine braids one can check that the Xεi all commute with each other. View the symbols εi as a basis of n so that

n=i=1n εi,and letL =i=1n εi. (1.5)

The affine braid group ,1,n contains a large abelian subgroup

X={XλλL}, (1.6)

where Xλ= (Xε1)λ1 (Xεn)λn for λ=λ1ε1++ λnεnL. The pole winding number of an affine braid b,1,n is κ(b) where κ:,1,n is the group homomorphism defined by κ(Xε1)=1 and κ(Ti)=0,2in. The affine braid group ,p,n is the subgroup of ,1,n of affine braids with pole winding number equal to 0 (mod p),

,p,n= { b,1,n κ(b)=0 (modp) } . (1.7)

Define

Q=i=2n (εi-εi-1) andLp=Q+ i=1npεi, (1.8)

for each nonnegative integer p. The lattice Lp is a lattice of index p in L. Then

,p,n= Xλ,Ti λLp,2in

and the group XLp= xλλLp is an abelian subgroup of ,p,n.

1.9 Affine Hecke algebras of type A

The affine Hecke algebra H,1,n (resp. H,p,n) is the quotient of the group algebra ,1,n (resp. ,p,n) by the relations

Ti2=(q-q-1) Ti+1,2in.

Let L and Xλ be as in (1.5) and (1.6). The subalgebra

[X]=span {XλλL} (resp.[XLp] =span{XλλLp} ) (1.10)

is a commutative subalgebra of H,1,n (resp. H,p,n). The symmetric group Sn acts on the lattice L by permuting the εi and the lattices Lp are Sn-invariant sublattices of L. Let si=(i,i-1)Sn and αi=εi-εi-1. For 2in and λL (resp λLp),

XλTi=Xsiλ Ti+(q-q-1) Xλ-Xsiλ 1-X-αi , (1.11)

as elements of H,1,n (resp H,p,n).

For each element wSn define Tw=Ti1Tip if w=si1sip is an expression of w as a product of simple reflections si such that p is minimal. The element Tw does not depend on the choice of the reduced expression of w [Bou1968, Ch. IV §2 Ex. 23]. The sets

{ XλTw λL,wSn } and { XλTw λLp,wSn }

are bases of H,1,n and H,p,n, respectively [Lus1989]. The center of H,1,n is

Z(H,1,n)= [ Xε1,, Xεn ] Sn =[X]Sn, (1.12)

and [XLp]Sn is the center of H,p,n (see Theorem 4.12 below).

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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